Local Indicators of Spatial Association-LISA

Transcription

1 Luc Amelin Local Indicators of Spatial Association-LISA The capabilities for uisualization, rapid data retrieual, and manipulation in geographic informution systems (GIS) haue created the need for new techniques of exploratory data analysis that focus on the spatial aspects of the data. The IcEentifcation of local patterns of spatial assocktion is an important concern in this respect. In this paper, I outline a new general class of local indicators of spatial association (LISA) and show how they allow for the decomposition of global indicators, such as Moran s I, into the contribution of each observation. The LISA statistics serve two purposes. On one hand, they may be interpreted as indicators of local pockets of nonstationarity, or hot spots, similar to the Gi and G: statistics of Getis and Ord (1992). On the other hand, they may be used to assess the influence of individual locations on the magnitude of the global statistic and to identifiy outliers, as in Anselin s Moran scatterplot (1993a). An initial eualuation of the properties of a LISA statistic is carried out for the local Moran, which is applied in a study of the spatial pattern of confict for African countries and in a number of Monte Carlo simulations. 1. INTRODUCTION The increased availability of large spatially referenced data sets and the sophisticated capabilities for visualization, rapid data retrieval, and manipulation in geographic information systems (GIS) have created a demand for new techniques for spatial data analysis of both an exploratory and a confirmatory nature (Anselin and Getis 1992; Openshaw 1993). Although many methods are available in the toolbox of the geographical analyst, only few of those are appropriate to deal explicitly with the spatial aspects in these large data sets (Anselin 1993b). In the analysis of spatial association, it has long been recognized that the as- The research of which this paper is an outgrowth was supported in part by grants SES (to the National Center for Geographic Information and Analysis, NCGIA) and SES from the U.S. National Science Foundation, and by grant GA-AS 9212 from the Rockefeller Foundation. Earlier versions were presented at the NCGIA Workshop on Exploratory Spatial Data Analysis and GIS, Santa Barbara, Calif., February 25-27, 1993, and at the GISDATA Specialist Meeting on GIs and Spatial Analysis, Amsterdam, The Netherlands, December 1-5,1993. The comments by Arthur Getis and two anonymous referees on an earlier draft are greatly appreciated. Luc Anselin is research professor of regional science at the Regionul Research Znstitute of West Virginia University, where he is also professor of economics, adjunct professor of geography, and adjunct professor of agricultural and resource economics. Geographical Analysis, Vol. 27, No. 2 (April Ohio State University Press Submitted 1/94. Revised version accepted 6/94.

2 94 / Geographical Analysis sumption of stationarity or structural stability over space may be highly unrealistic, especially when a large number of spatial observations are used. Spatial structural instability or spatial drift has been incorporated in a number of modeling approaches. For example, discrete spatial regimes are accounted for in spatial analysis of variance (Griffith 1978, 1992; Sokal et al. 1993), and in regression models with spatial structural change (Anselin 1988, 1990). Continuous variation over space is the basis for the spatial expansion paradigm (Casetti 1972, 1986; Jones and Casetti 1992) and spatial adaptive filtering (Foster and Gorr 1986; Gorr and Olligschlaeger 1994). In exploratory spatial data analysis (ESDA), the predominant approach to assess the degree of spatial association still ignores this potential instability, as it is based on global statistics such as Moran s I or Geary s c (as in Griffith 1993). A focus on local patterns of association (hot spots) and an allowance for local instabilities in overall spatial association has only recently been suggested as a more appropriate erspective, for example, in Getis and Ord (1992), Openshaw (1993), and Anse E n (1993b). Examples of techniques that reflect this approach are the various geographical analysis machines developed by Openshaw and associates (for exam le, Open- shaw, Brundson, and Charlton 1991; and Openshaw, Cross, and Char P ton 1990), the distance-based statistics of Getis and Ord (1992) (see also Ord and Getis 1994), and the Moran scatterplot (Anselin 1993a). Also, a few approaches have been suggested that are based on a geostatistical perspective, such as the pocket plot of Cressie (1991) and the interactive spatial graphics of Haslett et al. (1991). In the current paper, I elaborate upon this general idea and outline a class of local indicators of spatial association (LISA). These indicators allow for the decomposition of global indicators, such as Moran s I, into the contribution of each individual observation. I suggest that this class of indicators may become a useful addition to the toolbox of ESDA techniques in that two important interpretations are combined: the assessment of significant local spatial clustering around an individual location, similar to the interpretation of the Gi and G: statistics of Getis and Ord (1992); and the indication of pockets of spatial nonstationarity, or the suggestion of outliers or spatial regimes, similar to the use of the Moran scatterplot of Anselin (1993a). In the remainder of the paper, I first outline the general principles underlying a LISA statistic, and suggest how it may be interpreted. I next show how a number of familiar global spatial autocorrelation statistics may be expressed in the form of a LISA. As an example of a LISA, I examine the local Moran more closely, first empirically, comparing it to the G: statistic and the Moran scatterplot in an analysis of spatial pattern of conflict between African nations in the period This is followed by a series of simple Monte Car10 experiments, to provide further insight into the properties of the local Moran, its interpretation, and the relation between global and local spatial association. I close with some concluding remarks on future research directions. 2. LOCAL INDICATORS OF SPATIAL ASSOCIATION Definition As an operational definition, I su est that a local indicator of spatial association (LISA) is any statistic that satis 9 es the following two requirements: a. the LISA for each observation gives an indication of the extent of significant spatial clustering of similar values around that observation; b. the sum of LISAs for all observations is proportional to a global indicator of spatial association.

3 LUG Anselin / 95 More formally, but still in general terms, I express a LISA for a variable yi, observed at location a, as a statistic Li, such that where f is a function (possibly including additional parameters), and the y~~ are the values observed in the neighborhood Ji of i. The values y used in the computation of the statistic may be the original (raw) observations, or, more appropriately, some standardization of these in order to avoid scale dependence of the local indicators, similar to the practice often taken for global indicators of spatial association. For example, in Moran's I, as well as in its local version discussed in the next section, the observations are taken as deviations from their mean. The neighborhood Ji for each observation is defined in the usual fashion, and may be formalized by means of a spatial weights or contiguity matrix, W. The columns with nonzero elements in a given row of this matrix indicate the relevant neighbors for the observation that corresponds to the row, that is, the elements of Ji. Examples of criteria that could be used to define neighbors are first-order contiguity and critical distance thresholds. The spatial weights matrix may be row-standardized (such that its row elements sum to one) to facilitate inte retation of the statistics, but this is not required. However, when row stan ';p ardization is carried out, the function f(y6, y~,) typically corresponds to a form of weighted average of the values at all observations j E Ji. The Li should be such that it is possible to infer the statistical significance of the pattern of spatial association at location i. More formally, this requires the operationalization of a statement such as where Si is a critical value, and ai is a chosen significance or pseudo significance level, for example, as the result of a randomization test. The second requirement of a LISA, that is, its relation to a global statistic, may be stated formally as c Li = 74 (3) i where A is a global indicator of spatial association and 7 is a scale factor. In other words, the sum of the local indicators is proportional to a global indicator. For the latter, a statement such as Prob [A > S] 5 a, (4) indicates significant spatial association over the whole data set. Zdentijication of Local Spatial Clusters Local spatial clusters, sometimes referred to as hot spots, may be identified as those locations or sets of contiguous locations for which the LISA is significant. Similar to the rationale behind the significance tests for the Gi and Gt statistics of Getis and Ord (1992), the general LISA can be used as the basis for a test on the null hypothesis of no local spatial association. However, in contrast to what holds for the Gi and Gf statistics, general results on the distribution of a generic LISA may be hard to obtain. This is similar to the problems encountered in

4 96 / Geographical Analysis deriving distributions for global statistics, for which typically only approximate or asymptotic results are avai1able.l An alternative is the use of a conditional randomization or ermutation approach to yield empirical so-called pseudo significance levels P for example, as in Hubert 1987). The randomization is conditional in the sense that the value gi at a location a is held fixed (that is, not used in the permutation) and the remaining values are randomly permuted over the locations in the data set. For each of these resampled data sets, the value of Li can be computed. The resulting empirical distribution function provides the basis for a statement about the extremeness (or lack of extremeness) of the observed statistic, relative to (and conditional on) the values computed under the null hypothesis (the randomly permuted values). In practice, this is straightforward to implement, since for each location only as many values as there are in the neighborhood set need to be resampled. Note that this same approach can also easily be applied to the Gi and Gt statistics. A complicating factor in the assessment of significance of LISAs is that the statistics for individual locations will tend to be correlated, as pointed out by Ord and Getis (1994) in the context of their Gi and Gf statistics. In general, whenever the neighborhood sets Ji and Jk of two locations i and k contain common elements, the corresponding L, and Lk will be correlated. Due to this correlation, and the associated problem of multiple comparisons, the usual interpretation of significance will be flawed. Moreover, it is typically impossible to derive the exact marginal distribution of each statistic and the significance levels must be approximated b Bonferroni inequalities or following the approach suggested in Sid& ( I967r2 This means that when the overall significance associated with the multiple comparisons (correlated tests) is set to a, and there are m comparisons, then the individual significance aj should be set to either a/m (Bonferroni) or 1 - (1- (Sidn). The latter procedure, which yields slightly sharper bounds, is suggested by Ord and Getis (1994), with m = n, that is, the number of observation^.^ Note that the use of Bonferroni bounds may be too conservative for the LISA of individual locations. For example, if m is indeed taken to e ual the number of observations, then an overall significance of a = 0.05 woul 1 imply individual levels of ai = in a data set with one hundred observations, possibly revealing only very few if any significant locations. However, since the correlation between individual statistics is due to the common elements in the neighborhood sets, only for a small number of locations k will the statistics actually be correlated with an individual L,. For example, on a regular lattice using the ueen criterion of contiguity, first-order neighbors (ignoring border and comer ce ils) will have four common elements in their neighborhood sets, second-order neighbors three, and higher-order neighbors none. Clearly, the number of common neighbors does not change with the number of observations, so that using the latter in the computation of the Bonferroni bounds may be overly conservative. Hence, while it is obvious that some correction to the individual significance levels is needed, the extent to which it is indeed necessary to take m = n remains to be further investigated. With the exception of the results in Tiefelsdorf and Boots (1994), the general statement by Cliff and Ord (1981, p. 46) still holds: except for very small lattices, exact evaluation of the distribution function is impractical and approximations must be found. *An application of these procedures to the interpretation of the significance of a spatial correlogram was earlier suggested by Oden (1984). 3Note that the Sidik approach only holds when the statistics under consideration are multivariate normal, which is unlikely to be the case for the general class of LISA statistics [see also Savin (1980) for an extensive discussion of the relative merits of various notions of bounds].

5 Indication of Local Instability Luc Anselin / 97 The indication of local patterns of spatial association may be in line with a global indication, although this is not necessarily the case. In fact, it is quite possible that the local pattern is an aberration that the global indicator would not pick up, or it may be that a few local patterns run in the opposite direction of the global spatial trend. The second requirement in the definition of a LISA statistic is imposed to allow for the decomposition of a global statistic into its constituent parts. This is of interest to assess the extent to which the global statistic is representative of the average pattern of local association. If the underlying process is stable throughout the data, then one would expect the local indications to show little variation around their average. In other words, local values that are very different from the mean (or median) would indicate locations that contribute more than their expected share to the global statistic. These may be outliers or high leverage points and thus would invite closer scrutiny. This interpretation is roughly similar to the use of Cressie s (1991) pocket plots in geostatistics. By imposing the requirement that the Li sum to a magnitude that is proportional to a global statistic, their distribution around the mean ya/n can be evaluated. Extreme Li can be identified as outliers in this distribution, for example, as those values that are more than two standard deviations from the mean (the two-sigma rule) or more than 1.5 times the interquartile range larger than the third quartile (for example, in a box plot). This second interpretation of the LISA statistics is similar to the use of a Moran scatterplot to identify outliers and leverage points for Moran s I (Anselin 1993a). In general, it may be more appropriate than the interpretation of locations as hot spots suggested in the previous section when an indicator of global spatial association is significant. In this respect, it is important to note that the Getis-Ord Gi and G: statistics were suggested to detect significant spatial clustering at a local level when global statistics do not provide evideme of spatial association (Getis and Ord 1992, p. 201). Indeed, in their exam le, Getis and Ord find no global autocorrelation for SIDS cases in North CaroEna counties, while several significant local clusters are indicated. However, the opposite case often occurs as well, that is, a strong and significant indication of global spatial association may hide totally random subsets, particularly in large data sets. For example, in an analysis of the 1930 elections in Weimar Germany, OLoughlin, Flint, and Anselin (1994) found that a highly significant Moran s I at the level of 921 electoral districts in effect hides several distinct local patterns of spatial clustering and complete spatial randomness for six regional subsets. In such an instance, the distribution of the Li statistic as indicator of local spatial clustering will be affected by the presence of global spatial association. However, the second interpretation of LISA statistics, as indications of outliers or leverage points in the computation of a global statistic is not affected. I return to this issue in section LISA FORM OF FAMILIAR SPATIAL AUTOCORRELATION STATISTICS Local Gamma A broad class of spatial association statistics may be based on the general index of matrix association or r x index, originally outlined in Mantel (1967). The ap lication of the I index to spatial autocorrelation in a wide range of contexts is escribed in a series of papers by Hubert and Golledge (for example, Hubert 1985; Hubert, Golledge, and Costanzo 1981; Hubert et al. 1985; Costanzo,

6 98 / Geographical Analysis Hubert, and Golledge 1983).4 Such an index consists of the sum of the cross products of the matching elements a,j and bij in two matrices of similarity, say A and B, such that i j Measures of spatial association are obtained by expressing spatial similarity in one matrix (for example, a contiguity or spatial weights matrix) and value similarity in the other. Different measures of value similarity yield different indices for spatial association. For example, using aij = xixj yields a Moran-like meas- 2 ure, setting aij = (zi- xj) yields a Geary-like index, while taking ~j = 1zi - zjl results in an indicator equivalent to the one suggested by Royaltey, Astrachan, and Sokal (1975) [see, for example, Anselin (1986) for details on the implementation]. Since the I' index is a simple sum over the subscript i, a local Gamma index for a location i may be defined as Similar to what holds for the global I' measure, different measures of value similarity will yield different indices of local spatial association. It is easy to see that the I'i statistics sum to the global measure I?. It is possible that the distribution of the individual ri can be approximated using the principles outlined by Mielke (1979) and Costanm et al. (1983), though this is likely to be complex, and beyond the current scope. On the other hand, the implementation of a conditional permutation approach is straightforward. This allows the individual I'i to be interpreted as indicators of significant local spatial clusters. The second interpretation of the LISA statistic, as a diagnostic for outliers or leverage points can be carried out by comparing the distribution of the I'i to r/n. Local Moran As a special case of the local Gamma, a local Moran statistic for an observation i may be defined as where, analogous to the global Moran's I, the observations zi, zj are in deviations from the mean, and the summation over j is such that only neighboring values j E Ji are included. For ease of interpretation, the weights wij may be in rowstandardized form, though this is not necessary, and by convention, wii = 0. It can be easily seen that the corresponding global statistic is indeed the familiar Moran's I. The sum of local Morans is while Moran's I is 4Note that this statistic also forms the basis for the derivation of the distribution of Moran's I and Geary's c statistics in Cliff and (3rd (1981, p. 23 and chapter 2). In Getis (1991), this index is applied to integrate spatial association statistics and spatial interaction models into a common framework.

7 LUG Anselin / 99 i j i or where SO = Xi C. wi.. Using the same notation as ClifF and Ord (1981, p. 45), and taking m2 = gi 2/n as the second moment (a consistent, but not unbiased estimate of the variance), the factor of proportionality between the sum of the local and the global Moran is, in the notation of (3), y = som2. Note that for a row-standardized spatial weights matrix, SO = n, so that y = Xi 4, and for standardized variables (that is, with the mean subtracted and divided by the standard deviation), m2 = 1, so that 7 = So. Also, the same type of results obtain if instead of (7) each local indicator is divided by m2, which is a constant for all locations. In other words, the local Moran would then be computed as (11) The moments for Ii under the null hypothesis of no spatial association can be derived using the principles outlined by Cliff and Ord (1981, pp ) and a reasoning similar to the one by Getis and Ord (1992, pp ). For example, for a randomization hypothesis, the expected value turns out to be E[Ii] = -wi/(n - l), (13) with wi as the sum of the row elements, Cj wij, and the variance is found as Var [Ii] = q2)(n - bz)/(n - 1) +2Wi(kh)(2b2 - n)/(n - l>(n - 2) - wf/(n - I)', (14) with bz = ml/rme, m4 = xi zf/n as the fourth moment, wq4 = C.+ w:~, and ChZi 'Wikwih. The details of the derivation are given in appendix A. 2wqth) = cyi A test for significant local spatial association may be based on these moments, although the exact distribution of such a statistic is still unknown. This is further explored in section 5. Alternatively, a conditional randomization approach may be taken, as outlined earlier. Given the structure of the statistic in (12), it follows that only the quantity Cj w+j needs to be computed for each permutation (since the zi/m2 remains constant). Note that the randomization method applied to (12) will yield the same empirical reference distribution as when applied to the Getis and Ord Gi and G: statistics. Hence, inference based on this nonparametric approach will be identical for the two statistics. This easily follows from considering which elements in the statistics change for each permutation of the data. For example, the Gi statistic for an observation i is defined as

8 100 I Geographical Analysis where wij(d) are the elements in a distance-based weights matrix [for details, see Getis and Ord (1992)l. The only aspect of equation (15) that changes with each permutation is the numerator, since the denominator does not depend on the spatial allocation of observations. Clearly, this is the same term as the varying part of the numerator in (12). In other words, the pseudo significance levels (that is, the inference) generated with a permutation a proach applied to the Ii statistic will be identical to that for a Gi or G; statistic.! The interpretation of the local Moran as an indicator of local instability follows easily from the relation between local and global statistics expressed in equation (11). Specifically, the average of the Ii will equal the global I, up to a factor of proportionality. Extreme contributions may thus be identified by means of simple rules, such as the two-sigma rule, or by identifying outliers in a box plot. Note that this notion of extremeness does not imply that the corresponding Ii are significant in the sense outlined earlier, but only indicates the importance of observation i in determining the global statistic. This similarity to the identification of outliers, leverage and influence points in the Moran scatterplot (Anselin 1993a) will be further examined in the empirical illustration. Local Geay Using the same principles as before, a local Geary statistic for each observation i may be defined as 2 G = c wij (Zi- Zj), or as j using the same notation as before. Using expression (17) (without loss of generality), the summation of the q over all observations yields In comparison, Geary s familiar c statistic is r 1 Thus, the factor of proportionality between the sum of the local and the global Geary statistic is, in the notation of (3), y = 2nSo/(n - 1). (20) Clearly, for row-standardized weights, since So = n, this factor becomes 5See also Ord and Getis (1994) for a discussion of the relationship between their statistics and Moran s I.

9 Luc Anselin / 101 2n2/(n - 1). The q statistic is interpreted in the same way as the local Gamma and the local Moran. 4. ILLUSTRATION: SPATIAL PATTERNS OF CONFLICT IN AFRICA A geographical perspective has received much interest in recent years in the analysis of international interactions in general, and of international conflict in particular [see, for example, the review by Diehl (1992)l. Measures of spatial association, such as Moran s I, have been applied to quantitative indices for various types of conflicts and cooperation between nation-states, such as those contained in the COPDAB data base (Azar 1980). For such indices of international conflict and cooperation, both O Loughlin (1986) and Kirby and Ward (1987) found significant patterns of spatial association indicated by Moran s I. The importance of spatial effects in the statistical analysis of conflict and cooperation was confirmed in a study of the interactions among forty-two African nations, over the period , reported in a series of papers by O Loughlin and Anselin (O Loughlin and Anselin 1991, 1992; Anselin and O Loughlin 1990, 1992). For an index of total conflict in particular, there was strong evidence of both positive spatial autocorrelation (as indicated by Moran s I, by a I index of spatial association, and by the estimates in a mixed regressive, spatial autoregressive model), as well as of spatial heterogeneity in the form of two distinct spatial regimes (as indicated by Getis-Ord Gf statistics and the results of a spatial Chow test on the stability of regression coefficients). This phenomenon is thus particularly suited to illustrate the LISA statistics suggested in this paper. The illustration focuses on the two interpretations of the LISA statistics, as indicators of local spatial clusters and as diagnostics for local instability. It is approached from the perspective of exploratory spatial data analysis and the substantive interpretation of the models is not considered here [see O Loughlin and Anselin (1992) for a more extensive discussion]. The spatial pattern of the index for total conflict is illustrated in the quartile map in Figure 1, with the darkest shade corresponding to the highest quartile [for details on the data sources, see Anselin and O Loughlin (1992)l. The suggestion of spatial clustering of similar values that follows from a visual inspection of this map is confirmed b a strong positive and significant Moran s I of 0.417, with an associated stan d ard normal z-value of 4.35 (p < 0.001), and a Geary c index of 0.584, with associated standard normal z-value of (p < 0.002).6 These statistics are computed for a row-standardized spatial weights matrix based on first-order contiguity (common border), given the importance of borders in the study of international conflict (Diehl 1992). ldentijication of Local Spatial Clusters I first focus on a com arison of the identification of local spatial clusters provided by the Getis-Or x Gf statistic (as a standardized z-value) and the local Moran Ii indicator presented in equation (12). Note that the former, while being a statistic for local spatial association, is not a LISA in the terminology of section 2, since its individual components are not related to a global statistic of spatial association. This requirement is not needed for the identification of significant local spatial clusters, but it is im ortant for the second interpretation of a LISA, as a diagnostic of local insta! ility in measures of global spatial 6All computations were carried out with the Spacestat software for spatial data analysis (Anselin 1992); the map was created with the Idrist software (Eastman 1992), using the Spacestat-Iddsf interface: other graphics were produced by means of the SPlus statistical software.

10 102 / Geographical Analysis Q4 Missing Data 0 FIG. 1. Total Conflict Index for African Countries ( ) association (for example, in the presence of significant global association), which is discussed in the next section. Using the same row-standardized weights matrix as for the global measures given earlier, the results for the indicators of local spatial association are reported in the third and fifth columns of Table 1, for each of the forty-two countries in the example. The standardized z-value for Ii, computed by subtracting the expected value (13) and dividing by the standard deviation [the square root of (14)], is listed in the sixth column. Two indications of significance are given, one based on an approximation by the normal distribution, p, (in the seventh column of Table 1) and one derived from conditional randomization, using a sample of 10,000 permutations, p, (in the last column of Table l).7 As mentioned earlier, the pseudo significance obtained by means of a conditional randomization procedure is identical for G; and Ii. While this may suggest that the normal approximation shown to hold for G; (listed in column four) and assessed in detail in Ord and Getis (1994) may be valid for the Ii statistic as well, this has not been demonstrated. In fact, evidence from some initial Monte Carlo experiments in section 5 seems to indicate otherwise. Note that the two statistics measure different concepts of spatial association. For the G; statistic, a positive value indicates a spatial clustering of high values, and a ne ative value a s atial clustering of low values, while for the Ii, a positive value in ct 'cates spatial c lp ustering of similar values (either high or low), and negative values a clustering of dissimilar values (for example, a location with high values surrounded by neighbors with low values), as in the interpretation of the 7 More precisely, the sample consists of the original observed value of the statistic and the values computed for 9,999 conditionally randomized data sets.

11 Luc Anselin / 103 TABLE 1 Measures of Local Spatial Association Id country G: P 4 44) Pn R Gambia Mali Senegal Benin Mauritania Niger Ivory coast Guinea Burkina Faso Liberia Sierra Leone Ghana Togo Cameroon Nigeria Gabon CAR Chad Congo Zaire Angola Uganda oO O.oo00 Kenya Tanzania Burundi Rwanda Somalia Ethiopia Zambia Zimbabwe Malawi Mozambique South Africa Lesotho Botswana Swaziland Morocco Algeria Tunesia Libya Sudan O.oo O.oo00 Emt O.oo O.oo global Moran's I. This e lains the sign differences between the values in the third and fifth columns o ';p Table 1 (for example, for the first sixteen countries in the table). Following the suggestion by Ord and Getis (1994), a Bonferroni bounds procedure is used to assess significance. With an overall ar level of 0.05, the individual significance levels for each observation should be taken as 0.05/42, or Given this conservative procedure, the normal a proximation for both the GZ and the Ii show the same four countries to e XE ibit local *For a = 0.10, the corresponding individual significance level is Since normality was not demonstrated, the original Bonferroni bounds were used, rather than the slightly sharper Sidiik procedure suggested in Ord and Getis (1994). This does not affect the interpretation of the results in Table 1, since the difference between the two only appears at the fifth significant digit. For example, for a = 0.05, the Bonferroni bound is , while the Si& bounds are

12 104 / Geographical Analysis ean FIG. 2. Density of Randomized Local Moran for Uganda (i = 22) spatial clustering (with the significance levels in bold type in Table 1). They are Uganda (22), Kenya (23), Sudan (41), and Egypt (42), which themselves form a cluster in the northeast of Africa, part of the so-called Shatterbelt.9 This spatial clustering (or spatial autocorrelation) of both the Gf and the LISA statistics is a result of the way they are constructed, and should be kept in mind when visually interpreting a map of LISAs (or Gf). The conditional randomization approach provides a still more conservative picture of (pseudo) significant local spatial clustering, with only Sudan meeting the Bonferroni bound for an overall a = For this country, two out of the 9,999 statistics computed from the randomized samples exceed the observed one, clearly labeling the latter as extreme. lo Of the other three previously significant countries, only Kenya comes close to the threshold (with a seudo significance of ), but both Uganda (0.0031) and Egypt (0.0058) f ar 1 short of even the bounds for an overall (Y = Some insight into the reasons for the differences in interpretation between the normal ap roximation and the randomization strategy can be gained from Figure 2, whic ph shows the empirical distribution of the Ii for the 10,000 samples used in the computation of the pseudo significance for Uganda (22). This country was chosen since it has different significance indications between the two criteria, and it is not a boundary or comer location (it has five neighbors, which is about average for the sample). The density function in Figure 2 is smoothed, using a smoothing parameter of twice the interquartile distance. The sample average and the observed value are indicated on the figure (the latter with the label i = 22 ). The density under the curve for values larger than (the observed value) is , indicating its extremeness (but not significance according to the Bonferroni criterion). The distribution is clearly non-normal, and heavily skewed to the right (skewness is ). Its average of is smaller than the expected value under the null hypothesis for observation 22, which is In addition, its standard deviation of is more than 1.5 times the value that would be expected under the theoretical null distribution, or [the square root of expression (14)]. QThe identification numbers in parentheses correspond to the labels in the Moran scatterplot of Figure 3. lome Bonferroni bound for an overall significance level of a = 0.01 would be

13 LUG Anselin / 105 The differences between the empirical density in Figure 2 and (i) the theoretical moment and (ii) an approximation of the null distribution by a normal raise two important issues. First, the normal may not be an appropriate approximation, and higher order moments may have to be used, as in the approximation to the global I? statistics in Costanzo, Hubert, and Golledge (1983). However, it may also be that the sample size and/or the number of neighbors in this example (respectively, forty-two and five) are too small for a valid approximation by the normal.11 Secondly, and more importantly, the moments under the null hypothesis are derived assuming that each value is equally likely at any location, which is inappropriate in the presence of global spatial association. In other words, the theoretical moments in (13) and (14) do not reflect the latter. This is appropriate when the objective is to detect local spatial clusters in the absence of global spatial association [for example, as was the stated goal in Getis and Ord (1992)], but is not correct when global spatial association is present (as is the case in the example considered here). While the z-values for both G: and Ij would suffer from this problem, the conditional randomization strategy does not, since it treats the observations as $they were spatially uncorrelated. This issue is revisited in section 5. Indication of Local Instability The second interpretation of a LISA is as a diagnostic for outliers with respect to a measure of global association, in this example Moran s I. The Ii statistics are compared to the insights provided by the Moran scatterplot, suggested by Anselin (1993a) as a device to achieve a similar objective, that is, to visualize local instability in spatial autocorrelation. Note that the Moran scatterplot is not a LISA in the sense of this paper, since no indication of significant local spatial clustering is obtained. The principle behind the interpretation of the Moran scatterplot is that many statistics for global association are of the form z Az/dz, where z is a vector of observations (in deviations from the mean) and A is a matrix of known elements. In the case of Moran s I, the A is the row-standardized spatial weights matrix W. Given this form for the statistic, it may be visualized as the slope of a linear regression of Wz on z [see also Anselin (1980) for the interpretation of Moran s I as a regression coefficient]. A scatterplot of Wz on z [similar to a spatial lag scatterplot in geostatistics, for example, as in Cressie (1991)], with the linear regression line superimposed, provides insight into the extent to which individual (Wzi, zi) pairs influence the global measure, exert leverage, or may be interpreted as outliers, based on the extensive set of standard regression diagnostics (for example, Cook 1977; Hoaglin and Welsch 1978; Belsley, Kuh, and Welsch 1980). The Moran scatterplot for the African conflict data is given as Figure 3, with the individual countries labeled as in Table 1. The (Wzi, zi) pairs are given for standardized values, so that outliers may be easily visualized as points further than two units away from the origin. In Figure 3, both Sudan (41) and Egypt (42) have values for total conflict that are more than two standard deviations higher than the mean (on the horizontal axis of Figure 3), while Egypt also has values for the spatial lag that are twice the mean (vertical axis of Figure 3). The use of standardized values also allows the Moran scatterplots for different variables to be comparable. The four quadrants in Figure 3 correspond to the four types of spatial association. The lower left and upper right uadrants indicate spatial clustering of similar values: low values (that is, less t ll an the mean) in.see Getis and Ord (1992, pp ) for the importance of both sampie size and the number of neighbors for the normal approximation of the Gi and G: statistics.

14 106 / Geographical Analysis TOTCON FIG. 3. Moran Scatterplot for Total Conflict (I = 0.417) the lower left and high values in the upper right. Stated differently, the lower left pairs would correspond to negative values of the Gi and Gi, and the up er right pairs to positive values. With the Ii statistics, no distinction is possi YJ le between the two forms of association since both result in a positive sign. The upper left and lower right quadrants of Figure 3 indicate spatial association of dissimilar values: low values surrounded by high neighboring values for the former, and high values surrounded by low values for the latter. These correspond to Ii statistics with a negative sign. Since they are not cross-product statistics, the Gi and GI statistics do not capture this form of spatial association. While the overall pattern of spatial association is clearly positive, as indicated by the slope of the regression line (Moran s I), eleven observations show association between dissimilar values: eight in the upper left quadrant, also shown as light islands within the darkest clusters of Figure 1; and three in the lower right quadrant (Algeria, 38, Morocco, 37, and South Africa, 33), surrounded by countries in the first and second quartile in Figure 1. This may indicate the existence of different regimes of spatial association. The application of regression diagnostics for leverage to the scatterplot suggests that two observations deserve closer scrutiny. The highly significant local spatial association for Sudan (41) and Egypt (42) finds a match with the indication of leverage provided by the diagonal elements of the hat matrix. These are respectively (for Sudan) and (for Egypt), both distinctly larger than the usual cutoff of 2k/n (where Ic is the number of explanatory variables in the regression, or 2 in this example), or The third largest hat value of The diagonal elements of the hat matrix H = X(X X)- X, with X as the matrix of observations on the explanatory variables in a regression, are well known indicators of leverage. See, for example, Hoaglin and Welsch (1978).

15 Luc Anselin / FIG. 4. Local Moran Outliers for Uganda, 22, does not exceed this threshold. One may be tempted to conclude that the elimination of Egypt and Sudan from the sample would void the indication of spatial association, but this is not the case. Without both countries, Moran s I drops to 0.254, but its associated z-value is 2.53 (using the randomization null hypothesis), which is still highly significant at p < The distribution of Ii statistics for the sample can similarly be exploited to provide an indication of outliers or leverage points. In Figure 4, this is illustrated by means of a simple two-sigma rule. The mean of the distribution of the I* is Moran s I, or 0.417, and twice the standard deviation from the mean corresponds to the value of Clearly, this is exceeded by both Sudan (41), with a value of for I*, and by Egypt (42), with a value of While this is obviously not a test in a strict sense, it provides useful insight into the special nature of these two observations. All four indicators are in agreement in this respect, that is, the Ga and Ii as measures of local spatial clusters, and the Moran scatterplot and Ii as indicators of outliers. The substantive interpretation of the special nature of these observations is beyond the scope of the exploratory data analysis. The role of the latter is to point them out and by doing so to aid in the suggestion of possible explanations or hypotheses. Alternatively, the indication of strange observations may point to data quality problems, such as coding mistakes, or, in the case of spatial analysis, problems with the choice of the spatial weights matrix. 5. MONTE CARL0 EVIDENCE: GLOBAL AND LOCAL SPATIAL ASSOCIATION Two issues raised by the results of the empirical illustration in the previous section are revisited here by means of some initial Monte Carlo experiments. The first pertains to the distribution of the local Moran I* statistic under the

16 108 / Geographical Analysis null hypothesis of no (global) spatial autocorrelation. The second issue is the distribution of the local statistic when global spatial autocorrelation is present, and its implication for the assessment of significance. This may also have relevance for the distribution of the Gi and Gf statistics in this situation, since their distribution under the null is also based on the absence of global association. As pointed out earlier, it is quite common to study local association in the presence of global association, for example, this is the case in the illustration presented in the previous section. This second issue also has relevance for the assessment of outliers or local instability, that is, the second interpretation of the local Moran. It is well known that many spatial processes that produce s atially autocorrelated patterns also generate spatial heterogeneity. For examp P e, this is the case for the familiar spatial autoregressive process (Anselin 1990). The spatial heterogeneity indicated by LISAs, based on a null hypothesis of no spatial association may therefore be a natural characteristic of the spatial process, and not an indication of local pockets of nonstationarity. Two sets of experiments were carried out, one based on the same spatial weights matrix as for the African example (with n = 42), the other on the weights matrix for a 9 by 9 regular grid, using the queen notion of contiguity (with n = 81). Both weights matrices were used in row-standardized form. For each of these configurations, 10,000 random samples were generated with increasing degrees of spatial autocorrelation, constructed by means of a simple s atial autoregressive transformation. More formally, given a vector E of ran comly f generated standard normal variates, a spatially autocorrelated landscape was generated as a vector y: y = (I - pw)%, where p is the autoregressive parameter, taking values of 0.0, 0.3, 0.6, and 0.9, and I is a n by n identity matrix. While the resulting samples will be spatially autocorrelated for nonzero values of p, there is no one-to-one match between the value of p and the global Moran s I. As is well known, the latter is capable of detecting many different forms of spatial association, and is not linked to a specific spatial process as the sole alternative hypothesis. Distribution of the Local Moran under the Null Hypothesis The distribution of the standardized z-values that correspond to the Ii statistic was considered in detail for two selected observations, the location corresponding to Uganda, i = 22, for the African weights matrix, and the location corresponding to the central cell, i = 41, for the regular lattice. Not only are the dimensions of the data sets different in the two examples (n = 42 and n = 81), but also the number of neighbors differ for the observations under consideration, as they are respectively 5 and 8. The moments of each distribution for the z-values, based on the 10,000 replications, are given in the first row of Table 2. While the mean and standard deviation are rough1 in accordance with those for a standard normal distribution, the kurtosis an CI to a lesser extent the skewness are not. This is further illustrated by the density graph in Fi re 5 (for n = 81), which clearly shows the leptokurtic nature of the distri r ution and the associated thicker tails (compared to a normal density). The density graph for the African case is very similar and is not shown. Instead, a quantile-quantile plot for the African example is given in Figure 6, to further illustrate the lack of normality. While there is general agreement in the central section of the two distributions (total agreement would be shown as a perfect

17 Luc Anselin / 109 ~ TABLE 2 Moments of Local Moran with Global Spatial Autocorrelation* n = 42 n=81 P Mean St.Dev. Skew Kurtosis Mean St.Dev. Skew Kurtosis a. z-values for local Mom% 10,000 replications. using observation 22 for n = 42 and observation 41 for n = 81. linear fit), at the tails, that is, where it matters in terms of significance, this clearly is not the case. A more rigorous assessment of the distribution, based on an asymptotic chi-s uared test constructed around the third and fourth moments (Kiefer and Sa 9 mon 1983) strongly rejects the null hypothesis of normality in both cases. This more extensive assessment confirms (in a controlled setting) the earlier suggestion implied by the discrepancy between the significance levels under the normal approximation and the conditional randomization in Table 1. Note that the African example in Table 1 exhibited significant global spatial autocorrelation, while the simulations here do not (by design). Further results are needed to see whether larger sample sizes or higher numbers of neighbors are needed before normality is obtained. However, from the initial impressions gained here it would seem that the normal approximation may be inappropriate, and that higher moments (given the values for skewness and kurtosis in Table 2) would be needed in order to obtain a better approximation [for example, as in Costanzo, Hubert, and Golledge (1983) for the r statistic]. The implications of these results for inference in practice are that even when no global spatial autocorrelation is present, the significance levels indicated by a normal approximation will result in an over-rejection of the null hypothesis for a given Type I error. Clearly, a more conservative approach is warranted, although the exact nature of the corrections to the ai awaits further investigation. In the meantime, a conditional randomization approach provides a useful alternative. 2- x r 0 - x- I I I FIG. 5. Density of %-value for Local Moran (n = 81; 10,000 Replications)